Search results for "Normal"

p-Blocks relative to a character of a normal subgroup

2018

Abstract Let G be a finite group, let N ◃ G , and let θ ∈ Irr ( N ) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr ( G | θ ) relative to p. We call each member B θ of this partition a θ-block, and to each θ-block B θ we naturally associate a conjugacy class of p-subgroups of G / N , which we call the θ-defect groups of B θ . If N is trivial, then the θ-blocks are the Brauer p-blocks. Using θ-blocks, we can unify the Gluck–Wolf–Navarro–Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the Height Zero conjecture. We also prove that the k ( B ) -conjecture is true i…

On the product of a π-group and a π-decomposable group

2007

[EN] The main result in the paper states the following: Let π be a set of odd primes. Let the finite group G=AB be the product of a π -decomposable subgroup A=Oπ(A)×Oπ′(A) and a π -subgroup B . Then Oπ(A)⩽Oπ(G); equivalently the group G possesses Hall π -subgroups. In this case Oπ(A)B is a Hall π-subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G.

On finite groups generated by strongly cosubnormal subgroups

2003

[EN] Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in and, if Z is the hypercentre of G=, we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Thou…

Algebraically closed groups in locally finite group classes

1987

Existentially closed locally cofinite groups

1992

Let be a class of finite groups. Then a c-group shall be a topological group which has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups with -factor groups and trivial intersection. In this note we study groups which are existentially closed (e.c.) with respect to the class Lc of all direct limits of c-groups (where satisfies certain closure properties). We show that the so-called locally closed normal subgroups of an e.c. Lc-group are totally ordered via inclusion. Moreover it turns out that every ∀2-sentence, which is true for countable e.c. L-groups, also holds for e.c. Lc-groups. This allows it to transfer many known properties from e.c. L-group…

Primitive characters of subgroups ofM-groups

1995

One of the hardest areas in the Character Theory of Solvable Groups continues to be the monomial groups. A finite group is said to be an M-group (or monomial) if all of its irreducible characters are monomial, that is to say, induced from linear characters. Two are still the main problems on M-groups: are Hall subgroups of M groups monomial? Under certain oddness hypothesis, are normal subgroups of M-groups monomial? In both cases there is evidence that this could be the case: the primitive characters of the subgroups in question are the linear characters. This is the best result up to date ([4], [6]). Recently, some idea appears to be taking form. In [14], T. Okuyama proved that if G is an…

A note on character degrees ofp-groups and their normal subgroups

1992

NONVANISHING ELEMENTS FOR BRAUER CHARACTERS

2015

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zero…

Infinite factorized groups

1989

Normalities and Commutators

2010

We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and we define a new notion of normality in terms of this commutator. Our main result is to extend to any semi-abelian category the following well-known characterization of normal subgroups: a subobject K is normal in A if. and only if, {[A, K] <= K. (C) 2010 Elsevier Inc. All rights reserved.}